Unfortunately you may have some difficulties with your proposal. While your HL and BTL loan are separate loan facilities your mortgages are likely to cover "all sums due". I.e. Existing bank have the right to demand that you pay both mortgages off before they release the charge on your HL and they will probably insist on this. Also, the reduced payment arrangement on your BTL loan is likely in place to facilitate you meeting your full P&I repayments on the HL. There is no other logical reason why the bank would allow you this arrangement and it would likely cease if you tried to move your HL facility elsewhere.
You can certainly speak to them about the proposal but I doubt very much if they will let you go ahead with it.